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We define the \( (i, j) \)-liars’ domination number of \( G \), denoted by \( LR(i, j)(G) \), to be the minimum cardinality of a set \( L \subseteq V(G) \) such that detection devices placed at the vertices in \( L \) can precisely determine the set of intruder locations when there are between 1 and \( i \) intruders and at most \( j \) detection devices that might “lie”.
We also define the \( X(c_1, c_2, \ldots, c_t, \ldots) \)-domination number, denoted by \( \gamma _{X(c_1, c_2, \ldots, c_t, \ldots)}(G) \), to be the minimum cardinality of a set \( D \subseteq V(G) \) such that, if \( S \subseteq V(G) \) with \( |S| = k \), then \( |(\bigcup_{v \in S} N[v]) \cap D| \geq c_k \). Thus, \( D \) dominates each set of \( k \) vertices at least \( c_k \) times making \( \gamma_{X(c_1, c_2, \ldots, c_t, \ldots)}(G) \) a set-sized dominating parameter. We consider the relations between these set-sized dominating parameters and the liars’ dominating parameters.